Are long-term N-body simulations reliable?

Abstract

N-body integrations are used to model a wide range of astrophysical dynamics, but they suffer from errors which make their orbits diverge exponentially in time from the correct orbits. Over long time-scales, their reliability needs to be established. We address this reliability by running a three-body planetary system over about 200 e-folding times. Using nearby initial conditions, we can construct statistics of the long-term phase-space structure and compare to rough estimates of resonant widths of the system. Our statistics are approximately consistent for a wide range of numerical methods, including a Runge-Kutta method, Wisdom-Holman method, symplectic corrector methods, and a method by Laskar & Robutel. "Improving" an integrator did not affect the phase space accuracy, but simply increasing the number of initial conditions did.

On the reliability of simulations of collisional stellar systems

“…Thus, although an initial small difference can grow exponentially so that individual N -body models cannot reproduce the exactly same evolution of a real cluster, the statistical means are meaningful. A similar result in the orbits of planets is found in Hernandez et al (2020).…”

“…Therefore, Nesc of individual models is not reliable; to understand the properties of escapers in a low-mass star cluster, it is necessary to obtain ensemble statistics. Hernandez et al (2020) found a similar behaviour for the phase space structure of planetary dynamics.…”

“…It is much more important to have a proper algorithm to correctly treat the short-range interactions rather than achieving a good energy conservation by using an approximation (e.g., a softening potential). Hernandez et al (2020) found that the relevant metric for accuracy was the number of steps per bound orbit, rather than the energy error. They found that orbits needed at minimum about 16 steps per effective period at pericenter, a number which depends on eccentricity.…”

“…Portegies Zwart & Boekholt (2014) investigates how the accuracy of integration affects the few-body motions (the decay time) and found that small errors in individual simulations can finish at completely different orbits due to chaos, but the statistical mean of many repeating simulations can converge to the correct result. Hernandez et al (2020) investigated the long-term evolution of planetary systems. They found that the statistics of action variables are accurate as long as all orbits are properly resolved; e.g., 16 steps per effective pericenter period.…”

An algorithm for coalescence of classical objects and formation of planetary systems

“…So without fusion the angular momenta L I (t) and L G (t) with Newton's discrete dynamics are conserved separately, and at a fusion the total angular momentum is still conserved but with an exchange of angular momentum with δL I (t) = −δL G (t). The exact classical discrete dynamics with fusion of colliding objects can be used to explore the self-assembly at the emergence of planetary systems and to investigate the stability and chaotic behaviour of solar systems [26].…”